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Search Results: 1 - 10 of 3586 matches for " Abigail Thompson "
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Invariants of curves in RP^2 and R^2
Abigail Thompson
Mathematics , 2006, DOI: 10.2140/agt.2006.6.2175
Abstract: There is an elegant relation found by Fabricius-Bjerre [Math. Scand 40 (1977) 20--24] among the double tangent lines, crossings, inflections points, and cusps of a singular curve in the plane. We give a new generalization to singular curves in RP^2. We note that the quantities in the formula are naturally dual to each other in RP^2, and we give a new dual formula.
Finding geodesics in a triangulated 2-sphere
Abigail Thompson
Mathematics , 2014,
Abstract: Let S be a triangulated 2-sphere with fixed triangulation T. We apply the methods of thin position from knot theory to obtain a simple version of the three geodesics theorem for the 2-sphere [5]. In general these three geodesics may be unstable, corresponding, for example, to the three equators of an ellipsoid. Using a piece-wise linear approach, we show that we can usually find at least three stable geodesics.
Fibered knots and Property 2R
Martin Scharlemann,Abigail Thompson
Mathematics , 2009,
Abstract: It is shown, using sutured manifold theory, that if there are any 2-component counterexamples to the Generalized Property R Conjecture, then any knot of least genus among components of such counterexamples is not a fibered knot. The general question of what fibered knots might appear as a component of such a counterexample is further considered; much can be said about the monodromy of the fiber, particularly in the case in which the fiber is of genus two.
Unknotting tunnels and Seifert surfaces
Martin Scharlemann,Abigail Thompson
Mathematics , 2000,
Abstract: Let $K$ be a knot with an unknotting tunnel $\gamma$ and suppose that $K$ is not a 2-bridge knot. There is an invariant $\rho = p/q \in \mathbb{Q}/2 \mathbb{Z}$, $p$ odd, defined for the pair $(K, \gamma)$. The invariant $\rho$ has interesting geometric properties: It is often straightforward to calculate; e. g. for $K$ a torus knot and $\gamma$ an annulus-spanning arc, $\rho(K, \gamma) = 1$. Although $\rho$ is defined abstractly, it is naturally revealed when $K \cup \gamma$ is put in thin position. If $\rho \neq 1$ then there is a minimal genus Seifert surface $F$ for $K$ such that the tunnel $\gamma$ can be slid and isotoped to lie on $F$. One consequence: if $\rho(K, \gamma) \neq 1$ then $genus(K) > 1$. This confirms a conjecture of Goda and Teragaito for pairs $(K, \gamma)$ with $\rho(K, \gamma) \neq 1$.
Surfaces, submanifolds, and aligned Fox reimbedding in non-Haken 3-manifolds
Martin Scharlemann,Abigail Thompson
Mathematics , 2003,
Abstract: Understanding non-Haken 3-manifolds is central to many current endeavors in 3-manifold topology. We describe some results for closed orientable surfaces in non-Haken manifolds, and extend Fox's theorem for submanifolds of the 3-sphere to submanifolds of general non-Haken manifolds. In the case where the submanifold has connected boundary, we show also that the boundary-connected sum decomposition of the submanifold can be aligned with such a structure on the submanifold's complement.
Thinning genus two Heegaard spines in the 3-sphere
Martin Scharlemann,Abigail Thompson
Mathematics , 2001,
Abstract: We study trivalent graphs in $S^{3}$ whose closed complement is a genus two handlebody. We show that such a graph, when put in thin position, has a simple (i. e. non-loop) level edge.
Surgery on a knot in (Surface x I)
Martin Scharlemann,Abigail Thompson
Mathematics , 2008, DOI: 10.2140/agt.2009.9.1825
Abstract: Suppose F is a compact orientable surface, K is a knot in F x I, and N is the 3-manifold obtained by some non-trivial surgery on K. If F x {0} compresses in N, then there is an annulus in F x I with one end K and the other end an essential simple closed curve in F x {0}. Moreover, the end of the annulus at K determines the surgery slope. An application: suppose M is a compact orientable 3-manifold that fibers over the circle. If surgery on a knot K in M yields a reducible manifold, then either: the projection of K to S^1 has non-trivial winding number; or K lies in a ball; or K lies in a fiber; or K is a cabled knot.
On tunnel number one knots that are not (1,n)
Jesse Johnson,Abigail Thompson
Mathematics , 2006,
Abstract: We show that the bridge number of a $t$ bridge knot in $S^3$ with respect to an unknotted genus $t$ surface is bounded below by a function of the distance of the Heegaard splitting induced by the $t$ bridges. It follows that for any natural number $n$, there is a tunnel number one knot in $S^3$ that is not $(1,n)$.
On the additivity of knot width
Martin Scharlemann,Abigail Thompson
Mathematics , 2004,
Abstract: It has been conjectured that the geometric invariant of knots in 3-space called the width is nearly additive. That is, letting w(K) in N denote the width of a knot K in S^3, the conjecture is that w(K # K') = w(K) + w(K') - 2. We give an example of a knot K_1 so that for K_2 any 2-bridge knot, it appears that w(K_1 # K_2) = w(K_1), contradicting the conjecture.
Knots and k-width
Joel Hass,J. Hyam Rubinstein,Abigail Thompson
Mathematics , 2006,
Abstract: We investigate several integer invariants of curves in 3-space. We demonstrate relationships of these invariants to crossing number and to total curvature.
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